Let be the unknown number. Say the first trial gives the value . Then we can conclude that is between

trial 2 gives a result of . Again: is between . since we have information from the previous round about the interval in which could be, intersect the two intervals.

repeat.

But will this always terminate with a conclusion " must be..."??
No, since the number k could come up every single round and no new information is gained. If we require that a new number be given every single round then the worst case scenario is around rounds, since every integer in the interval might come up.

An interesting question is, what is the expected number of steps required for you to be 10% or 50% or 95% sure of the number?